MCQ: Cone - 2 - SSC CGL MCQ

The slant height of a right circular cone is 13 cm and the area of the base is 144π cm2.

Concept used:

Volume of a cone = (π x R2 x H)/3
Slant height 
where R = radius of the base, H = Height
The area of a circle = πR² (R being the radius of the circle)

Calculation:

Let the radius of the base of the right circular cone be R cm.
According to the question,
πR² = 144π
⇒ R² = 122
⇒ R = 12 (since radius can't be negative)
Height of the right circular cone, H

⇒ 5 cm(since height can't be negative)

Now, the volume of the cone

⇒ (π x 122 x 5)/3

⇒ 240π cm³

∴ The volume of the cone is 240π cm³.

Volume of cone = πr2h/3

Volume of cone

⇒ [1/3] x π x r² x h

⇒ [1/3] x [22/7] x 7 x 7 x 15

⇒ 22 x 7 x 5

⇒ 770 cm³

Given

The circumference of the base = 33 cm
Height solid cone = 16 cm

Calculation:

Circumference of the base of a cone = 2πr

⇒ 2πr = 33

⇒ 2 x (22/7) × r = 33
⇒ r = 21/4

Volume of cone = (1/3) πr²h

⇒ (1/3) x (22/7) x (21/4) x (21/4) × 16

⇒ 21 x 22 = 462 cm³

∴ The volume of the cone is 462 cm³.

Given:

Curved surface area of a cone = 4664 cm²
radius = 28 cm

Formula used:

Curved surface area of cone = π × r × l

where l = slant height; r = radius 

Calculation:

Curved surface area = 4664
⇒ π × r × l = 4664
⇒ 22/7 × 28 × l = 4664 
⇒ l = 53 cm
∴ Slant height i.e. l is 53 cm.

Given:

Slant height of cone l = 30 cm

Radius of cone r = 14 cm

Formula:

Total surface area of cone = πr (l + r)

Calculation:

Total surface area of cone
⇒ πr(r + l)
⇒ [22/7] x 14 x (14 + 30)
⇒ 22 x 2 x 44
⇒ 1936 cm²

Given:

Radius of the bottom of the frustum =  12 cm.
Radius of the top of the frustum = 6 cm.
Height = 8 cm

Formula used:
Slant Height of a Frustum of a cone, 
Curved Surface Area of a Frustum of a cone, CSA = π × L(R + r)
Where Radius of the bottom of the frustum =  R, Radius of the top of the frustum = r, Height = H, and Slant Height = L

Calculation:
Hence, the slant height of a Frustum of a cone, L 

Thus, the curved surface area of a Frustum of a cone, CSA
⇒ π x 10 x (12 + 6)
⇒ π x 10 x 18
⇒ 180π cm²

∴ The curved surface area of the frustum is 180π cm².

Given,

Circumference of base of cone = 132
Radius of cone = 132 × 7/22 × 1/2 = 21 cm
Then,
Slant height of cone = 21 + 8 = 29 cm
Using Pythagoras theorem,
⇒ (Slant height)² = (Height)² + (Radius)²
⇒ Height² = 29² – 21²
⇒ Height = 20 cm
Volume of cone = 1/3 × πr²h
= 1/3 × 22/7 × 21 × 21 × 20
= 9240 cm³

Formula of curved surface area of a cone = πrl

where r is the radius and l is the slant height of cone.

⇒ r = √(l² – h²)

⇒ r = √(256 – 207)

⇒ r = √49 = 7 cm

∴ Curved surface area of the cone = (22/7)x 7 x 16 = 352 cm²

Formula used:

Volume of frustum of cone 

Where r₁ and r₂ are radius of the frustum of cone

Given:

Radius (r₁) of the upper base = 6/2 = 3 cm

Radius (r₂) of lower the base = 4/2 = 2 cm

Height = 14 cm

Calculation:

∴ The capacity of the glass = 88.67π cm³

Given:

Radius of the base (R) = 5 cm 
Radius of the top (r) = 3 cm
Height (H) of frustum = 6 cm

Formula used:

Volume of Frustum (V) = 1/3 πH (R² + Rr + r²)

Calculations:

According to the question,

Volume of Frustum = 1/3 π × 6 × [(5)² + (5 × 3) + (3)²]
⇒ V = π x2 x [25 + 15 + 9]
⇒ V = π x 2 x [49] = 98π cm³ 
∴ The volume of frustum is 98π cm³.